Abstract
In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then, a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.
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References
Carmi, S., Turgeman, L., Barkai, E.: On distributions of functionals of anomalous diffusion paths. J. Stat. Phys. 141(6), 1071–1092 (2010)
Chen, M., Deng, W.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37(2), A890–A917 (2015)
Chen, M., Deng, W.H.: Discretized fractional substantial calculus. ESAIM Math. Model. Numer. Anal. 49(2), 373–394 (2015)
Chen, S., Shen, J., Zhang, Z., Zhou, Z.: A spectrally accurate approximation to subdiffusion equations using the log orthogonal functions. SIAM J. Sci. Comput. 42(2), A849–A877 (2020)
Deng, W., Hou, R., Wang, W., Xu, P.: Modeling anomalous diffusion: from statistics to mathematics. World Scientific (2020)
Deng, W.H., Chen, M.H., Barkai, E.: Numerical algorithms for the forward and backward fractional Feynman-Kac equations. J. Sci. Comput. 62(3), 718–746 (2015)
Deng, W.H., Li, B., Qian, Z., Wang, H.: Time discretization of a tempered fractional Feynman-Kac equation with measure data. SIAM J. Numer. Anal. 56(6), 3249–3275 (2018)
Fan, E., Li, C., Stynes, M.: Discretised general fractional derivative. Math. Comput. Simulation 208, 501–534 (2023)
Gunzburger, M., Li, B., Wang, J.: Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise. Numer. Math. 141(4), 1043–1077 (2019)
Gunzburger, M., Wang, J.: A second-order Crank-Nicolson method for time-fractional PDEs. Int. J. Numer. Anal. Model. 16(2), 225–239 (2019)
Hao, Z., Cao, W., Lin, G.: A second-order difference scheme for the time fractional substantial diffusion equation. J. Comput. Appl. Math. 313, 54–69 (2017)
Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Engrg. 346, 332–358 (2019)
Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39(6), A3129–A3152 (2017)
Jin, B., Li, B., Zhou, Z.: An analysis of the Crank-Nicolson method for subdiffusion. IMA J. Numer. Anal. 38(1), 518–541 (2018)
Li, B., Luo, H., Xie, X.: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. SIAM J. Numer. Anal. 57(2), 779–798 (2019)
Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B 24(4), 1989–2015 (2019)
Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383–406 (2012)
Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)
Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)
Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65(213), 1–17 (1996)
Mustapha, K., Abdallah, B., Furati, K.M.: A discontinuous Petrov-Galerkin method for time-fractional diffusion equations. SIAM J. Numer. Anal. 52(5), 2512–2529 (2014)
Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)
Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)
Sun, J., Nie, D., Deng, W.: Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data. J. Sci. Comput. 84(1), Paper No. 6, 23 (2020)
Sun, J., Nie, D., Deng, W.: High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data. Appl. Numer. Math. 161, 82–100 (2021)
Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximation for solving space fractional diffusion equations. Math. Comp. 84(294), 1703–1727 (2015)
Turgeman, L., Carmi, S., Barkai, E.: Fractional Feynman-Kac equation for non-Brownian functionals. Phys. Rev. Lett. 103(19), 190201 (2009)
Wang, Y., Yan, Y., Yan, Y., Pani, A.K.: Higher order time stepping methods for subdiffusion problems based on weighted and shifted Grünwald-Letnikov formulae with nonsmooth data. J. Sci. Comput. 83(3), Paper No. 40, 29 (2020)
Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56(1), 210–227 (2018)
Zhang, Y.N., Sun, Z.Z., Liao, H.L.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)
Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J. Sci. Comput. 37(2), A701–A724 (2015)
Zhou, H., Tian, W.Y.: Two time-stepping schemes for sub-diffusion equations with singular source terms. J. Sci. Comput. 92(2), Paper No. 70, 28 (2022)
Zhou, H., Tian, W.Y.: Crank-Nicolson schemes for sub-diffusion equations with nonsingular and singular source terms in time. J. Sci. Comput. 98(2), Paper No. 50, 24 (2024)
Acknowledgements
The authors are very grateful to the two anonymous referees for their careful reading of the original manuscript and their valuable suggestions, which greatly improved the quality of this paper. WYT thanks Dr. Han Zhou for helpful discussions.
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This work was partially supported by the National Natural Science Foundation of China grants 12071343 and 11701416.
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Communicated by: Bangti Jin
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Hao, L., Tian, W. Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data. Adv Comput Math 50, 87 (2024). https://doi.org/10.1007/s10444-024-10188-7
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DOI: https://doi.org/10.1007/s10444-024-10188-7
Keywords
- Backward fractional Feynman-Kac equation
- Fractional substantial derivative
- Weighted and shifted Grünwald difference formula